QR Decomposition Interpretation

Question

What is exact relationship between matrix R and input matrix A in QR factorization? Say, R gives the structure of A or R is a representation of A. How? We have Q'A = R. Does it mean A is projected to the subspcae of Q? If so, is R a representation of A in another space?

Answer 1

One way to compute the QR decomposition is by Givens rotations, which means that $Q$ can be observed as a composition of many 2D rotations ("many" being at most $n(n-1)/2$ in case of a square $A$ of order $n$).

Now, $Q^T$ (or $Q'$, as you denote the transpose) is just an inverse of those rotations, which are the same rotations applied in the opposite order and direction.

Geometrically, this means that when you apply the rotations of $Q^T$ to the columns of $A$, the first column will fall on the first coordinate axes (so, it will have all the coordinates, except maybe the first one, equal to zero), the second column will fall into the space spanned by the first two coordinate axes (having at most first two coordinates not equal to zero), etc.

Try to envision it on a 3D object: pick an edge, rotate it to an x-axis, then pick another edge, rotate it to xy-plane without moving the first one away from the x-axis, and that's your QR on the columns defined by those two edges (any additional edges will end up somewhere in 3D space, so no zeros for them, except maybe by a happy accident).

P.S. Note that this is different from a projection which, unless it is identity, drops your dimension which loses you information. Like flattening an object instead of rotating it.

Answer 2

$Q$ corresponds to an orthonormal basis constructed from the original $A$ basis using the Gram-Schmidt process.

You can see $R$ as a change of basis, from $Q$ to $A$. Its triangular shape is explained by the fact that vectors from $A$ are incorporated one at a time.

Answer 3

It's eccentric to describe QR this way, but what if you want to think of it your matrix as a linear function rather than a fixed set of vectors? You can think of $M=QR$ as decomposing an operator $x \rightarrow Mx$ into a stabilizer (R) of a given flag (the ordered standard basis) followed by an isometry (Q). Flags and stabilizers lurk in other areas of numerical linear algebra, such as Krylov subspaces.